Monte‐Carlo calculation of output correction factors for ionization chambers, solid‐state detectors, and EBT3 film in small fields of high‐energy photons

Abstract High‐energy accelerators are often used in oncological practice, but the information on the small‐field dosimetry for the photon beams with nominal energy above 10 MV is limited. The goal of the present work was to determine the values of the output correction factor (kQclin,Qreffclin,fref) for solid‐state detectors (Diode E, PTW 60017; microDiamond, PTW 60019), EBT3 film, and ionization chambers (Semiflex, PTW 31010; Semiflex 3D, PTW 31021; PinPoint, PTW 31015; PinPoint 3D, PTW 31016) in the small fields formed by 10, 15, 18, and 20 MV photon beams. The output correction factors were calculated by Monte‐Carlo method using EGSnrc toolkit for six field sizes (from 0.5×0.5cm2 to 10×10cm2) for isocentric and constant source‐to‐surface distance (SSD) techniques. The decrease in the field size led to an increase in kQclin,Qreffclin,fref for ionization chambers, while for solid‐state detectors and radiochromic film, kQclin,Qreffclin,fref were less than unity at the smallest field size. A larger sensitive volume of ionization chamber corresponded to a stronger deviation of output correction factor from unity: 1.847 (125 mm3 PTW 31010) versus up to 1.183 (16 mm3 PTW 31016) at the smallest field of 10 MV beam. The calculated output correction factors were used to correct the output factors for PTW 60017, PTW 60019, and EBT3. The deviation of the corrected output factor from the results of Monte‐Carlo simulation did not exceed 3% in the fields from 1.0×1.0cm2 to 4.0×4.0cm2 for 10 and 18 MV beams. Thus, Diode E, microDiamond, and EBT3 film can be recommended for small‐field dosimetry of high‐energy photons.

beam axis; (2) there is partial occlusion of the primary photon source by the collimating devices on the beam axis; (3) the size of the detector is similar or large compared to the beam dimensions. Small-field dosimetry has several technical features: differences in photon energy spectra 5,6 and positioning accuracy of the detectors compared to traditional fields, absence of electronic equilibrium, 7-10 as well as volume averaging effect. [11][12][13] In addition,placing the detector in a small field can cause a noticeable perturbation in the photon beam. 14,15 All these subtleties of small-field dosimetry were first summarized by Alfonso et al., 16 and in 2017, International Atomic Energy Agency, together with American Association of Physicists in Medicine, published a detailed report TRS-483 Code of Practice, 4 containing the state of art of the dosimetry of static small fields for maximum photon beam energies up to 10 MV (mainly for 6 MV beams). At the same time, the information on the small-field dosimetry for the photon energies above 10 MV is extremely limited, although medical accelerators generating bremsstrahlung photons with energies up to 20-24 MV could be used in oncological practice. 17,18 For example, the energies up to 18-20 MV are utilized in some cases in the dynamic conformal arc SBRT. 19,20 To measure the absorbed dose in small fields, it is important to take into account that each detector has its own individual characteristics, which may affect the dosimetry accuracy. For example, corrections for the dose profiles calculation as well as beam quality correction factors for the case of small fields of high bremsstrahlung energies were calculated in Refs. [21][22][23][24] In addition, the measured output factors for small fields require applying of correction factors for the ratio of the detector readings. 16 Determination of output factors and output correction factors for photon energies ≥10 MV was performed in the works of Refs. 1,[25][26][27] and Refs., 28,29 respectively. In the present paper, the values of the output correction factor were determined in the small fields of 10, 15, 18, and 20 MV photon beams for a set of detectors widely used in clinical dosimetry: a diode detector Dosimetry Diode E (type 60017), a diamond detector microDiamond (type 60019), as well as an EBT3 dosimetric film, and four models of microionization chambers: a Semiflex Chamber (type 31010), a Semiflex 3D Chamber (type 31021), a PinPoint Chamber (type 31015), and a PinPoint 3D Chamber (type 31016).

Output correction factor for detectors in small fields
The new formalism for small-field dosimetry recommended in TRS-483 4 is based on the establishment of the new type of reference field for beam calibration, machine-specific reference field (msr-field), f msr . For most medical accelerators, the msr-field matches with the standard reference field,f msr ≡ f ref = 10 × 10 cm 2 . For those accelerators, which cannot form a standard 10 × 10 cm 2 reference field, the closest to 10 × 10 cm 2 field is taken as the msr-field. For relative dosimetry of small f clin -beams, it is necessary to strictly apply the experimental definition of the output factor as the ratio of doses in f clin and f ref fields 4 :

Monte-Carlo calculation of phase space files
In the present paper, the output correction factors were calculated by Monte-Carlo method using EGSnrc software toolkit. 30 At the first step, the phase space files (PSFs) of the radiation field containing complete information on the characteristics of photons crossing virtual planes were calculated using BEAMnrc user code. The geometry of the head of Varian accelerators (valid for Novalis TX, Trilogy, Clinac iX, DX, C/D, EX, and cX), which included the main elements such as targets, primary collimator, flattening filters, mirror, ion camber, secondary collimator, was set in the input file BEAMnrc according to the package of data of the Varian High-Energy Clinac machines. 31 The baseline electron source parameterization was mono-energetic with a circularly symmetric Gaussian FWHM = 1 mm (for 10 MeV) and FWHM = 1.4 mm (for 15, 18, and 20 MeV). 32 The PSFs were generated on the surface of a water phantom at the distance of 90 and 100 cm from the accelerator's target, which corresponds to isocentric irradiation and irradiation with fixed solid-state detector, respectively. The collimation jaws were set so that the radiation field sizes at d = 10 cm depth in the first case and on the surface of the water phantom in the second case were 0.5 × 0.5 cm 2 , 1.0 × 1.0 cm 2 , 2.0 × 2.0 cm 2 , 3.0 × 3.0 cm 2 , 4.0 × 4.0 cm 2 , and 10 × 10 cm 2 . The number of particles in each PSF was at least 2 × 10 6 photons∕cm 2 .
The following values of the parameters in BEAMnrc user code were used to simulate the radiation transport through the accelerator head (here and below they were selected empirically based on the recommendations presented in NRCC Report PIRS-701 30 to obtain optimal statistics and calculation time): electron and photon cutoffs ECUT = 0.7 MeV, PCUT = 0.01 MeV; maximum energy-loss fraction per step ESTEPE = 0.25; energy threshold ESAVE = 2.0 MeV. To increase the number of bremsstrahlung, the technique of their directional splitting in the target was used with the parameter value NBRSPL = 2000 for 15 and 18 MV photon beams and NBRSPL = 500 for 10 and 20 MV beams.

Experimental verification of Monte-Carlo calculation
The dose profiles (at d = 10 cm depth) and percentage depth doses (PDDs) were calculated using DOSXYZnrc toolkit and measured experimentally in the water phantom for different sizes of the small field at SSD = 90 cm. The PSFs calculated by the method described previously were used as the radiation sources for the calculation. The following values of the parameters in the user code were used: ECUT = 0.7 MeV, PCUT = 0.01 MeV, ESTEPE = 0.25, ESAVE = 2 MeV, and nsplit = 10. The absorbed dose was determined on a uniform grid in rectangular voxels with Δx = Δy values from 0.025 to 0.25 cm and Δz values from 0.1 to 0.25 cm depending on the field size. The stories number was selected so as to obtain a statistical uncertainty of no more than 1.0%.
As an example, Figure 1a,c shows the results of the calculation of dose profiles and their experimen-tal measurement using microDiamond and PFD 3G (IBA, Germany) detectors for 0.5 × 0.5 cm 2 , 1.0 × 1.0 cm 2 , 2.0 × 2.0 cm 2 , and 3.0 × 3.0 cm 2 fields of 10 MV photon beam. Comparison of central-axial dose distributions for 2.0 × 2.0 cm 2 , 3.0 × 3.0 cm 2 , and 10 × 10 cm 2 fields, measured with the same detectors and simulated by Monte-Carlo method, is presented in Figure 1b,d. It is clearly seen that the measured and simulated data are in good agreement. For both detectors, the discrepancy between theoretical and experimental dose profiles within the high-dose region reached 5% for the smallest studied field, while for the other field sizes, it was less than 2%. In the penumbra region, the best agreement was found for the diamond detector. For PFD 3G , the mean distance-to-agreement was ≤ 0.15 mm for all field sizes. When comparing the PDDs, the discrepancies in the readings of the studied detectors and the data of Monte-Carlo simulation did not exceed 2%. So, the measured and simulated data are in good agreement. The verification of PSFs was also carried out for 18 and 20 MV beams (using Varian Clinac iX and Varian Trilogy, respectively). The results were generally similar in accuracy to the case of 10 MV (data not shown).

Calculation of equivalent square field size
The values of full width at half maximum (FWHM) for inline and cross-line directions of the dose profile (FWHM x and FWHM y ), as well as the equivalent square field size S clin were determined in the water phantom at a 10 cm depth according to Ref. 4 by Monte-Carlo simulation. The S clin values corresponding to different beam energies and nominal field sizes (S col ) formed by the collimation jaws for SSD = 90 and 100 cm are presented in Table 1.

Monte-Carlo calculation of output correction factor
The output correction factors were calculated by Monte-Carlo method using Equation (2) for two solid-state detectors (Dosimetry Diode E, microDiamond) and four models of micro-ionization chambers (Semiflex Chamber, Semiflex 3D Chamber, PinPoint Chamber, PinPoint 3D Chamber) manufactured by PTW (Germany), as well as for Gafchromic EBT3 film (Ashland, USA). The specification of PTW detectors were obtained from the manufacturer's manual, 33 while the radiochromic film specification was taken from Ref. 34 Their characteristics are presented in Table 2. The models of the detectors adapted from EGSnrc are schematically shown (not in scale) in Figure 2a. An insignificant simplification was made for simulation of the ionization chambers: the radii of the stems were taken equal to the radii of the chambers. Since the design of solid-state detectors is F I G U R E 1 Comparison of dose profiles (a and c) and PDDs (b and d) for various field sizes calculated theoretically (lines) and determined experimentally (circles): (a and b) the data for microDiamond detector (c and d) the data for PFD 3G detector.   much more complex, 35 the simplifications were used for the simulation of their sensitive elements according to Refs. 36,37 The sensitive volumes were set as cylinders with dimensions and volumes specified in the manufacturer's manual. As it was noted by Francescon et al., 38 possible difference in the real size of the sensitive volume of microDiamond detector does not greatly affect the results of their simulation. The radiochromic film was set as 1 cm radius sample, and the dimensions of the active volume where the absorbed dose was calculated were r = 0.1 cm and d = 30¯m. The detectors were placed in the 40 × 40 × 50 cm 3 water phantom so that their reference points were located on a geometric axis of the beams. The axes of the ionization chambers were perpendicular to the beam axis (Figure 2b), and the reference points of the chambers were located at a depth of d = 10 cm. The solid-state detectors were placed in parallel, and the radiochromic film was placed perpendicular to the beam axis so that the centers of their sensitive volumes were located at a 10 cm depth. To determine the absorbed dose in water, a cylinder of 0.2 cm height (h) and 0.2 cm radius (r) was set at a depth of 10 cm for 10 × 10 cm 2 field. The cylinder of h = 0.1 cm and r = 0.05 cm was set for 0.5 × 0.5 cm 2 field, and the cylinders of h = 0.2 cm and r = 0.1 cm were used for the other field sizes. Note that the larger sizes of cylinder were used for larger field sizes to improve the statistics. The absorbed dose was scored in rectangular voxels with a side of 0.1-0.4 cm depending on the field size, centered in the point of interest on the beam axis.

TA B L E 1 Values of equivalent field size lengths (S clin ).
The absorbed doses in water and in the detectors were calculated using DOSXYZnrc and egs_chamber applications, respectively. The following parameters were used in the simulation: AE = AP = ECUT = ESAVE = 10 keV. The electron range rejection (ESAVE When calculating the absorbed dose for ionization chambers, the type A statistical uncertainty was ≤0.3%, for solid-state detectors and radiochromic film, it was ≤(0.2-0.9)%. The statistical uncertainty was <0.2% when calculating the absorbed dose in water. Moreover, despite the known sensitivity of the output factor simulation results to the size of the electron focal spot and the geometry of the accelerator head, 39,40 a good agreement between calculated dose profiles and ones measured experimentally was established in our case.

Determination of output factors
The output factors for microDiamond, Dosimetry Diode E, PinPoint 3D Chamber, and EBT3 radiochromic film were determined experimentally for TrueBeam and Clinac IX accelerators (Varian Medical System, USA). The output factors were obtained by Equation (1). For this, the average reading of the detectors by three measurements were taken. For all detectors except for the radiochromic film, the output factors were measured in a MP3 Water Phantom System (PTW) using Unidose E electrometer (PTW). The detectors were placed in the center of the beam based on the measurement of two orthogonal profiles. For solid-state detectors, the voltage was set to 0 V in accordance with the manufacturer's recommendations, and for the ionization chamber, the voltage was of 400 V, respectively. No additional corrections for polarization and recombination coefficients were applied. For EBT3, the output factors were measured in polymethyl-methacrylate (PMMA)/Solid Water phantoms at the same conditions. Before the measurements, the films were calibrated against the dose measured with 0.6 cm 3 Farmer ionization chamber (type 30010; PTW) according to TRS-398. 2 The films were scanned using Expression 10000 XL scanner (Epson, Japan) within 24 h after exposure. Image processing was performed using ImageJ software according to the technique described in Ref. 41 The output factors were also calculated for the case of water using Monte-Carlo method. The values of Ω

Monte-Carlo calculation of output correction factor
The output correction factors for ionization chambers, solid-state detectors, and radiochromic film upon exposure with 10, 15, 18, and 20 MV photon beams in different irradiation techniques calculated using Monte-Carlo method are presented in Tables 3-6, respectively. The values obtained for isocentric irradiation technique slightly varied from those for constant SSD technique due to the ≈10% difference in the equivalent field size length (see Table 1). The ionization chambers and the solid-state detectors showed opposite dependencies of the output correction factor on the radiation field size. For the ionization chambers, the output correction factor increased with a decrease in the field size, while for diode, diamond detector, and radiochromic film, the unity for solid-state detectors and radiochromic film in small radiation fields was less than 4%, while the readings of ionization chambers differed more significantly. For 3.0 × 3.0 cm 2 and 2.0 × 2.0 cm 2 fields, the reading deviation was ≤4%, while for the fields less than 2.0 × 2.0 cm 2 , the deviation was up to ≈85%. The readings of the ionization chambers in small fields correlated with their sensitive volumes: a larger sensitive volume corresponded to a stronger deviation from unity with decreasing field size. For example, the highest calculated value of k f clin ,f ref Q clin ,Q ref under 10 MV exposure for 125 mm 3 Semiflex Chamber was 1.847, while for 16 mm 3 PinPoint 3D Chamber, the highest value was 1.183. Since the detectors with the deviation of the output correction factor from unity larger than 5% are not recommended for clinical dosimetry, 4 the obtained data prove some limitations of the studied ionization chambers for small-field dosimetry in the high-energy region.

Experimental measurements of output factor of the clinical fields
As an example, comparison of corrected values of the output factors of microDiamond, Dosimetry Diode E, PinPoint 3D Chamber, and EBT3 film with Monte-Carlo simulation for 10 and 18 MV photon beams are presented in Tables 7 and 8, respectively. We did not use other ionization chambers because according to the simulation results, they turned out to be unsuitable for dosimetry of small fields for the case of high-energy photon beams. According to Equation (1), the ratios of the detectors' readings in the clinical and reference fields were multiplied by the previously determined values of the output correction factor from Tables 3 and 5. For microDiamond, the deviation of the corrected values detector from the results of computer simulation did not exceed 1%-2%. The corrected values of the output factor for Dosimetry Diode E coincided with the results of Monte-Carlo calculation within 2%-3%. The values matching with these data of 98%-99% were also found for EBT3 radiochromic film. The uncertainty of experimental and calculated data was ≈ 1.5% (1 ). The values of the output factors calculated by Monte-Carlo method were in good agreement with the data obtained for Exradin A16 Ion Chamber (0.007 cm 3 sensitive volume; Standard Imaging, USA) for the case of Varian machine in Ref. 1

DISCUSSION
The    beams (especially for the fields less than 3.0 × 3.0 cm 2 ). First, this is due to an increase in the size of the region of absence of transverse electronic equilibrium; second, due to the ionization chamber volume averaging effects; and moreover, due to an increase in the slope of the decay of the photon beam transverse profile and an increase in the radiation flux perturbations caused by the placing the detectors within the field. As expected, the output correction factor for the field sizes less than 2.0 × 2.0 cm 2 strongly depends on the sensitive volume of the ionization chambers. If, in accordance with TRS-483, the difference of ≈5% is taken as the applicability limit of the ionization chamber for smallfield dosimetry, 4 all studied ionization chambers can be used for the field sizes above 2.0 × 2.0 cm 2 and Pin-Point 3D for the field sizes above 1.0 × 1.0 cm 2 . These measurement results correspond to the recommendations of these ionization chambers manufacturer. 33 In addition, for the case of 10 MV beam, the deviations of our results for Semiflex and PinPoint 3D Chambers from corresponding TRS-483 4 data ranged from 0.2% to 0.7%.
The dependence of the output correction factors on the field size for the solid-state detectors and the radiochromic films differs significantly from that obtained for the ionization chambers. For the Dosimetry Diode E, this can be explained, first, by the volumetric effect due to the extremely small size of the sensitive volume; second, by the close to the absolute visibility of the radiation source from the sensitive volume; and third, by the material of the sensitive volume of the detector. An increase in the contribution of scattered radiation to the fluence caused by a decrease in the field size leads to a change in the ratio of the effective mass energy absorption coefficients of water and silicon. For Diode E, the largest discrepancy with TRS-483 4 equals to 3% was found for the smallest studied field of 0.5 × 0.5 cm 2 . For the other fields, the deviations were 0.3%-1.1%.
The EBT3 radiochromic film had the closest to unity values of the output correction factor in small fields among all the detectors under study. This was largely due to the closeness of the effective atomic number and density of the active layer of the film to those of water parameters. According to Butson et al., 34 the effective atomic number of the EBT3 active layer is Z eff = 7.46, while effective atomic number of water is Z eff = 7.42. In addition, the placing of the dosimetric film into water practically does not disturb the distribution of radiation fluxes due to its small thickness. According to the results of Casar et al., 27 the volume averaging coefficient of the dosimetric film for 6 and 10 MV photon beams is k vol = 1.002.
There are very contradictory data in the literature for the case of a diamond detector. As noted in Ref., 42 in various publications, the agreement in the character of the dependence of k f clin ,f ref Q clin ,Q ref on the field size for small radiation fields with a side of ≥1 cm is observed for 60 Co, 6 and 10 MV beams, whereas for smaller field sizes, the data differ significantly. In Ref., 27 the publications on this topic were divided into three groups. In the first two groups, which included works with Monte-Carlo calculations (numerical approach) and semi-empirical approach (combines measurements and numerical calculations), an upward deviation of the values of the output correction factor from unity was found. 43,44 In the third group, which included experimental works (empirical approach), the values of k f clin, f ref Q clin, Q ref were found less than unity, as in Refs. 28,45 The same result for a diamond detector was obtained for 6 and 10 MV beams in Ref. 27 However, in the high-energy region (>10 MV), this point is not studied yet.
In the present work, we observed a decrease in k f clin, f ref Q clin, Q ref with decreasing field size for the diamond detector under 10 MV photon irradiation. The increase in the beam quality did not change the character of the dependence within the uncertainty of results.The values of output correction factor obtained for microDiamond in the present work match with the data by Casar et al. 27 with an accuracy from 98.5% to 99.9% in case of 10 MV beam of Varian TrueBeam linac. And for case of Elekta Versa HD, the convergence of the data was about 97.3%-99.5% (except 0.5 × 0.5 cm 2 field-in this case, the difference was ≈7.4%). For the photon beams ≥10 MV, the deviation in the values of the output correction factor from unity for a diamond detector in our case was ≈2%-4% for all field sizes. This correlates well with the data by Casar et al. 27 for Varian TrueBeam, while the deviation for the case of Elekta Versa HD accelerator in their work reached ≈10% (for the smallest field). Thus, in absolute values,the output correction factors obtained in this work deviate by 0%-0.7% from the TRS-483 data for 10 MV machines. 4 Thus, although the atomic number of carbon (Z C = 6) is less than that of water, for studied photon energies, a higher dose response was observed relative to water. This effect may be caused by the fact that the secondary charged particles generated in the materials adjacent to the sensitive volume of the detector make the main contribution to ionization due to the extremely small sensitive volume of the diamond detector.

CONCLUSION
In the present work, the values of the output correction factor for a set of detectors widely used in clinical small-field dosimetry were calculated by Monte-Carlo method with use of EGSnrc software toolkit in the little studied region of photon energy above 10 MV. The obtained data demonstrate the limitations of the studied ionization chambers (Semiflex Chamber, PTW 31010; Semiflex 3D Chamber, PTW 31021; PinPoint Chamber, PTW 31015; PinPoint 3D Chamber, PTW 31016) for small-field dosimetry in the high-energy region. It is found that in small fields formed by 10-20 MV photons, the dependence of the output correction factors on the field size for solid-state detectors and radiochromic film differs crucially from that for the ionization chambers. The deviation of the output correction factors from unity for solid-state detectors and EBT3 film was less than 5%. Also, for solid-state detectors and EBT3 film, the experimental output factors, adjusted by the calculated values of the output correction factors, were in good agreement with the results of Monte-Carlo simulation within the uncertainty. Thus, Diode E (PTW 60017), microDiamond (PTW 60019), and EBT3 film have shown acceptable results and can be recommended to small-field dosimetry. All studied ionization chambers can be used in a more limited range of the field size (above 2.0 × 2.0 cm 2 ), and PinPoint 3D can be used for the field size above 1.0 × 1.0 cm 2 .